Wednesday, 25 July 2012

Unit - 8 Methods and Teaching aids

METHODS AND TECHNIQUES OF TEACHING MATHEMATICS

For teaching of every subject a
method is very important. It is nothing but a scientific way of the
presentation of the subject, keeping in view the psychological and physical
requirements of the students. It is through method only that it is possible to
make a subject interesting and useful. Without a method, teaching would be
pretty hard and difficult. The selection of a suitable method depends upon the
objectives of the lesson, needs of the learner and nature of the content.

The following are the methods
which are suitable for classroom teaching of mathematics.

Lecture method

Inductive and Deductive method.

Analytic and Synthetic method

Problem Solving method

Project method

Heuristic method

Laboratory method

Lecture method

The ‘Lecture method’ is otherwise
known as ‘the chalk and talk method’ where in verbal explanation is given for
facts. This is a teacher centered method, teacher is an active participant and
child is a passive learner. It is a one way traffic or flow of ideas. This is
the method of imparting information through a speech. This is a one man’s show
because the children remain passive through out the process. This method is
most commonly used in colleges and not a very suitable method for teaching
mathematics at high school classes. In this method the students are provided
with readymade information by the teacher. The teacher goes ahead with the
subject matter at his own speed.

Advantages ofLecture method

It is an
easy, concise and attractive method. Using this method the teacher feels safe
and secure.

Using this
method, a large amount of subject matter can be presented within a short time
andthe prescribed syllabus can be
covered easily.

It can be used for a large number of students.

This
method gives the students as well as the teacher a sense of satisfaction and
achievement.

Lecture
method trains good listeners who are able to concentrate on subject matter for
a long duration.

This
method suits the teacher. It is convenient for him because he has not to give
individual help to the students.

It is
economical with respect to time and money.

Limitations

This is a teacher- centered method. So this is
against psychological principles.

It does not provide opportunities to develop
various mental abilities like reasoning, logical thinking etc.

In this method students become passive
recipients of information as their involvement in classroom interaction is
negligible.

Receiving information is not mathematics
learning and hence it does not enhancemathematical ability of the students.

In this method there is no way to ensure the
students concentration and understanding of the subject matter presented to
them.

It does not provide for individual differences
and individual needs.

This method does not help in developing
problem-solving skills.

It does not provide for corrective feed back and
remedial help to slow learners.

It does not call for the development of mental
faculties such as power of observation, reasoning, critical thinking,
independent thinking and so on.

Uses

It can be used

while introducing a new
concept.

for explaining abstract
concepts.

for giving demonstration.

for summing up and reviewing
certain concepts.

while initiating a discussion.

Inductive method

Inductive method is based on
induction. Induction is the process of providing a universal truth or a theorem
by showing that it is true of any particular case, it is true for the next case
in the same serial order and hence true for any such cases. So the technique of
making transition from particular facts to generalizations about these facts is
known as the process of induction. Thus it is a method of arriving at a formula
or a rule by observing a sufficient number of particular instances. If one rule
applies to a particular case and is equally applicable to different similar
cases, it is accepted as a general rule or formula. Therefore, in this method
we proceed from particular to general, from concrete instances to abstract
rules and from simple examples to complex formula. A formula or a
generalization is arrived at through inductive reasoning.

This
method has been found to be very suitable for teaching of mathematics because
many mathematical formulae and generalizations are the results of induction.

Example

Angle sum property in a triangle

When a student measures the
angles of several triangles and finds that in each case the sum of the angles
approximates to 1800, he has the background to generalize that sum
of the interior angles in a triangle is 1800.

Establishing the formula (a + b)2
= a2 + 2ab + b2

Students may be asked to find out
the square value in each of the cases like (x + y)2, (m +n)2
, (p +q)2etc by the simple
method of multiplication.After doing
these different multiplications, they may be helped in generalizing that ( 1st
term + 2 nd term )2 = (1 st term)2
+ (2 nd term)2 + 2 (1 st term) (2 nd
term).

Steps in Inductive method

Selection of
a number of cases.

Observation
of the cases under given conditions.

Finding
common relations.

Arriving at
generalization.

Verification
or application.

Advantages

It
is a logical method and develops critical thinking.

It
encourages active participation of the students in learning.

It
provides ample opportunities for exploration and observation.

It
sustains the interest of students asthey proceed from knownto
unknown.

It
enhances self-confidence.

It
discourages cramming.

It
helps in increasing the pupil- teacher contact.

Limitations

This
method is limited in range and is not suitable for all topics.

This
method is not suitable for higher classes because higher order mathematical
principles cannot be generalized through the observation of concrete cases.

It
is time consuming and laborious method.

Inductive
reasoning is not absolutely conclusive because the generalization made with the
help of a few specific examples may not hold good in all cases.

The
ability and capacity of problem solving cannot be developedby the use of this method.

Uses

Inductive
method is most suitable where

Rules
are to be formulated

Definitions
are to be formulated

Formulae
are to be derived

Generalizations
of laws are to be arrived at

It
can be used in lower classes where certain simple generalizations are to be
made.

Deductive method

Deductive method is the opposite
of the inductive method. It is based on deduction. Deductive reasoning is the
process of drawing logical inferences from established facts or fundamental
assumptions. In this methodwe proceed
from general to particular and from abstract to concrete. The students are
asked to apply these rules to solve more problems. This approach is not
suitable for exploration.

In
this method, we begin with the formula or rule or generalization and apply it
to a particular case.

Example-

(a + b)2 = a2
+ 2ab + b2

Find (3 + 2)2

By applying the identity,

(3 + 2)2 = 32
+ 2 . 3 . 2 + 22

=
9 + 12 + 4

=
25

Steps in deductive method

Clear recognition of the problem: A clear recognition of the
problem statement provides the basis link for thinking process and the solution
to the problem.

Search for a tentative hypothesis

Search for a tentative
hypothesis, a tentative solution to the problem.

Formulation of a tentative hypothesis

Search for the solution leads to
the formulation of a tentative hypothesis that appears to have promise as a
possible or probable solution to the problem. The tentative hypothesis has its
basis on certain axioms or postulates or propositions or rules and formulae
that have been accepted to be true.

Verification

Finally the hypothesis that has
been formulated is to be verified as the right solution to the problem at hand.

Advantages

It saves time and labour for both
the teacher and the student.

It enhances speed, skill and
efficiency in solving problems.

It is a short and elegant method.

It helps in fixation of formulae
and rules as it provides adequate opportunities for practice and revision.

This method suits all types of
students.

It helps in increasing the memory
power of the students, as the students are required to memorize a large number
of laws, formulae etc.

Limitations

It is not a psychological method.

It encourages cramming.

It is not suitable for beginners.

It does not encourage students
involvement in learning.

It is not suitable for the
development of thinking, reasoning and discovery.]

Since it is based on blind
memory, once the formula or rule is forgotten, it is not possible to rediscover
them.

Students get disinterested as
they are only passive listeners.

Uses

This method is suitable for
fixation and retention of facts and rules.

It is suitable for teaching
demonstrative geometry.

This method is suitable for
average students.

This method provides skill, speed
and efficiency in solving problems.

Inductive and Deductive Method

Inductive method

Deductive method

Based on inductive reasoning

Based on deductive reasoning

Particular to General

General to Particular

A psychological method

An unpsychological method

A method of discovery that
stimulates intellectual powers

A method of presentation that
does not develop originality and creativity

Emphasis is on reasoning

Emphasis is on memory

Encourages meaningful learning

Encourages rote learning

Most suitable for initial
stages of learning

Suitable for practice and
application

Suitable for lower classes

Most suitable for higher
classes

Enhances active participation
of the students

Makes the student passive
recipient ofknowledge

Lengthy, time consuming and
labourious

Short, concise and elegant

Facilitates discovery of rules
and generalizations

Enhanced speed, skill and efficiency
in solving problems

It gives opportunity for the
better pupil teacher relationship

In this method the relationship
remains nominal

Analytic method

The word ‘analytic’ is derived
from the word ‘analysis’ which means ‘breaking up’ or resolving a thing into
its constituent elements. This method is based on analysis and therefore, in
this method we break up the problem in hand into its constituent parts so that
is ultimately gets connected with something obvious or already known. In this
process we start with what is to be found out (Unknown) and then think of
further steps and possibilities which may connect with the known and find out
the desired result. Hence in this method we proceed from unknown to known

Example

If a/b = c/dP.T ac + 3b2=c2 + 3bd

bcdc

To prove this using analytic
method, begins from the unknown

The unknown isac + 3b2=c2 + 3bd

bcdc

ac + 3b2=c2 + 3bdis true

bcdc

Ifac + 3b2=c2 + 3bdis true

b
d

If d( ac + 3b2)=b(
c2 + 3bd)

If dac + 3b2d= bc2 + 3b2d

If dac = bc2

If
da = bc

Ifa/b = c/dwhich is given to be true.

ac + 3b2=c2 + 3bd

bcdc

Advantages

It helps in developing reasoning power of the
students

It leaves no doubt in the minds of the students
as every step is justified.

It facilitates clear understanding of the
subject matter as every step is derived by the student himself.

It is a psychological method.

No cramming is required in this method

Students take active role in the learning
process resulting in longer retention and easier recall of what they learn.

It develops self-confidence in the pupil.

Limitations

It is a time consuming and lengthy method. So it
is uneconomical.

This method may not be suitable for all topics
of mathematics.

In this method facts are not presented in a neat
and systematic order.

Uses

This method is suitable for teaching arithmetic,
algebra and geometry.

This method is suitable for solving complicated
problems

Synthetic method

Synthetic
method is derived from the word synthesis. Synthesis is the complement of
analysis. To synthesis isto combine the
constituent elements to produce something new. In this method we start with
something already known and connect it with the unknown part of the statement.
Therefore, in this method one proceeds from known to unknown. It is the process
of combining known bits of information to reach the point where unknown
information becomes obvious and true.

The usual form of statements of proofs
found in textbook are examples of synthetic method. Beginning with known
definitions, assumptions and axioms, the sequence of steps are deducted and
conclusions are arrived at.

Ex

If a/b =
c/dP.T ac + 2b2=c2 + 2bd

bccd

a/b = c/d(given)

Add 2b/c to both sides (Why? It is not explained)

a/b + 2b/c=c/d+ 2b/c

ac + 2b2=c2 + 2bd( It was to be proved)

bccd

Advantages

This method is logical as in this method one
proceeds from the known to unknown

It saves time and labour

It suits majority of students

It can be applied to majority of topics in
teaching of mathematics

It is a neat method in which we present the
facts in a systematic way.

It is short and elegant

Limitations

It leaves many doubts in the minds of the
learner and offers no explanation for them

It makes the students passive listeners and
encourages cramming

This is an unpsychological method

In this method there is a scope for forgetting

There is no scope for discovery and enquiry

It does not provide full understanding

If the student forgets the sequence of steps, it
would be very difficult to reconstruct the proof/solution.

Uses

This method can be used by an
average teacher and also suits for the average student

This method is suited for the
final presentation of proofs of theorems and solutions to problems in a logical
and systematic manner

Comparison of Analytic and Synthetic methods

Analytic method

Synthetic method

Here one proceeds from unknown
to known

Here one proceeds from known to
unknown

Here one starts from what is to
be proved and then

comes back to what is given

Here one starts from what is
given and ends with what is to be proved

Analysis means breaking up into
components

Synthesis means combining the
elements to get something new

Lengthy, laborious and time
consuming

Short, concise and elegant

Encourages meaningful learning

Encourages rote learning

Easy to discover

Once forgotten, not easy to
recall

Valid reasons to justify every
step in the sequence

No justification for every step
in the sequence

Application of inductive
reasoning

Application of deductive
reasoning

Active participation of the
learner

Learner is a passive listener

A psychological method

A logical method

Problem solving method

The problem solving method is one
which involves the use of the process of problem solving or reflective thinking
or reasoning. It may be noted that all problem solving doesn’t necessarily include
reflective thinking or reasoning because problem may be solved through trial
and error or analogy.

Problem
solving method, as the name indicates, begins with the statement of a problem
that challenges the students to find a solution. In the process of solving the
problem the students may be required to gather data, analyse and interpret
information to arrive at solution to the problems.

According
to James Ross ‘Problem solving is an educational device where by the teacher
and the pupils attempt in a conscious, planned, purposeful manner to arrive at
an explanation or solution to some educationally significant difficulty

Steps in problem solving

1. Identifying and defining the problem

Teacher has to select problematic
situation and motivate children to ask questions. Then the children have put a
number of questions. Then the problem is identified. The identified problem
should be based on the requirement of children their competency and
intellectual level. The students should be able to identify and clearly define
the problem.

2. Analysing the problem

The problem should be carefully
analysed as to what is given and what is to be found out. Given facts must be
identified and expressed, if necessary in symbolic form.

3. Formulating tentative hypothesis

The focus at this stage is
hypothesizing searching for a tentative solution to the problem. Analysis of
the given data and analysis of interrelationship among the given facts helps
the students in formulating hypothesis or educated guesses as the solution to the
problem at hand.

4. Testing the hypothesis

Appropriate methods should be
selected to test the validity of tentative hypothesis as a solution to the
problem. If it is not proved to be the solution, the students are asked to
formulate alternate hypothesis and proceed.

5. Checking the result or verification of the result

At this step the students are
asked to determine if their results substantiate the expected solution

Advantages

Problem solving
provides a real life experience to the children

It
develops in pupils good habits of planning, thinking, reasoning and independent
work.

It
develops initiative and self-responsibility among the students

It takes
in to account individual differences

It helps
the students to develop reflective thinking

It helps
the students to approach future problems with confidence

This
method is helpful in the development of harmonious relationship between the
teacher and taught

It helps
the children develop mental traits of open-mindedness and tolerance as the
children see many sides to a problem and listen to many points of view.

Limitations

Not all the students are problem
solvers.

It cannot be suited to large
crowded classed

It is time consuming and
consequently it is not possible to cover the syllabus on time

Project method is developed and
applied practically by Dr.Kilpatrick. It is the outcome of the pragmatism-
ideas of Sir John Dewey. “ What is to taught should have a direct relationship
with the actual happenings in life”, this central idea forms the basis of
project method.

According
to Dr. Kilpatrick ‘A project is a unit of whole hearted purposeful activity
carried on preferably in its natural settings. Ballard defined project ‘as a
bit of real life that has been imparted in to school’. All definitions stated
above emphasis the project should be purposeful activity related to life and it
should be carried out in a natural environment. In project method, teaching and
learning are considered from the child’s point of view and in this method
knowledge and skills are learnt by pupils through practical handling of problem
in their natural setting. This method is an ideal way of promoting creativity,
arousing curiosity and inculcating the spirit of enquiry among the students

Basic principles of project method

The project method is based on
the psychological principles of learning namely

The project should arise out of
the felt needs of the students. The teacher should provide such situations to
students which may arouse some suitable questions to which the students seek
answer. It should look important, must be interesting and purposeful for the
students. The teacher can provide a variety of situations through discussion,
questioning, library work, field work etc

Selecting and purposing

The selection of the project is
done by the students themselves. The teachershould refrain from purposing any project; otherwise the whole purpose
of the method would be defeated. How ever the teacher can guide the students in
the selection of a good project, keeping in mind the interest, aptitude and
ability of the students. In this step the nature and goal of the project is
clearly determined as well as the limits and scope of the project is clearly
defined.

Planning

For any successful project
planning is essential. Planning involves the selection of the unit appropriate
and feasible set of activities to be executed. The students should choose the
most practical plan of action. The students themselves should do the planning
with the teacher as a guide. While planning the points to be taken in to
consideration are (i) The degree of complexity of the project. (ii) time
allotted to finish the project and (iii) availability of material resources.
Discussion may be held among students before the final draft of the plan is
agreed upon.

Execution

In this step the teacher helps
the students in assigning work to different students in accordance with their
interest, aptitude and capabilities. Each member of the group should be
actively involved in the execution of the project. The teacher should carefully
supervise and guide the students in the execution of the project as per the
proposed action plan.

Evaluation

The students along with the
teacher should review the progress of the project at frequent intervals. This
is to ensure that the students are progressing towards the realization of the
objectives of the project. Without evaluation, the project can’t move out of
focus. The evaluation of the project has to be done in the light of

The students are required to
maintain a complete record of work including the choice of the project, the
planning, the discussions held and duties assigned. Also reference and books
consulted and readings taken, difficulties faced, guidance sought, details of
places visited and surveyed and so on should be carefully recorded.

Advantages

It is based on sound
psychological principles

It provides scope for independent
work and individual development

It provides for individual
differences as the students can select the activity and work at their own pace.

It promotes social interaction,
inculcates spirit of co-operation and exchanges of experiences among the
students

It provides opportunities for
children to acquire a lot of skills- observation, reference and
interpretation

In this method children are
active participants in the learning task

It develops self-confidence and
self-discipline among the students

It upholds the dignity of labour

It arouses interest of the pupils

The facts are easily remembered

It makes the learning more
interesting and facilitates better understanding of the subject matter as the
learning is related to reality and the world around him.

Limitations

The project method is un
economical in terms of time and not possible to fit in to the regular time
table

It does not provide any training
in mathematical thinking and reasoning

The learning is incomplete and
uniform learning or balanced learning is not possible for all students as each
student perform a different activity

For the success of this method
the teachers should be exceptionally resourceful and gifted and knowledgeable

It is an expensive method as it
makes use of a lot of resources which are not immediately available in the
school

Syllabus cannot be completed on
time using this method

Uses

Project method can be used to
arouse interest and encourage initiative and to provide a sense of achievement
to the students

Project method provides a
practical approach to learning

Heuristic method

This method involves finding out
by the pupil, instead of merely telling of everything by the teacher. This
method demands complete self activity or self-education on the part of the
learner

The
term ‘heuristic’ is derived from the Greek word ‘Heurisco’ which means ‘I
discover’. This method was advocated by Professor H.E Armstrong, who felt that
by placing the student in the position of a discoverer, he would learn much
more than being merely told about things. In this method the teacher becomes on
looker and the child tries to move ahead independently with out any help or
guidance. This method makes the student self reliant and independent. The
teacher’s job is not to solve problems for the pupil, but to enable the student
to solve problems for himself. It aims at developing a scientific or heuristic
attitude in the pupil

Ex:-

The population of a city is
25,000. If the percentage increase in population is 2% per year, then find out
the population of the city after 2 years.

Solution

Teacher will tell the students to
repeat this question many times and will tell them to find the solution with
the help of suitable research questions and he will encourage them for finding
correct solution. We can understand the solution of the problem with the help
of questions asked by the teacher.

The questions asked by the
teacher and expected answers give by the students are listed in the following
table

Questions

Expected answers

What has to be calculating in
the given question?

Population of the city after 2
years

How can we find out this?

First find the population after
one year

What is the rate of increase in
population?

2%per year

How can we find population at
the end of first year?

Students can find that increase
in population at the end of first year = 25,000 X 2/100 = 500

In this way what will be the
total population at the end of first year

Students can tell after solving
that population at the end of first year = 25,000 + 500 = 25,500

Population increase for II year
can be calculated on which figure?

Students tell that population
increase for II year can be calculated on 25,500

How can we calculate this
increase?

Students can solve by their
efforts. Population increases at the end of II year = 25,500 X 2/100 = 510

Now how to calculate the
population at the end of II year?

Students will be able to
understand that population of 2 years will be 25,500 + 510 = 26,010

In this way solution to the given
problems can be found by Heuristic method.

Advantages

It develops the habit of enquiry and
investigation among the students

It is a psychological method, knowledge is not
thrown upon the child rather he is made to discover things himself.

It develops self-confidence, self-reliance and
scientific attitude

It provides for individual differences as each
student can work at his own pace.

This method demands quite alertness and presence
of mind on the part of the student for responding to the heuristic questions
asked by the teacher.

It helps in the development of social skills as
the students have to co-operate with one another.

Limitations

This
method is feasible only with a highly resourceful teacher and a small class

It is
lengthy, time consuming method and hence it is difficult to cover the
prescribed syllabus in time.

It is not
suitable for beginners as the students need enough guidance at early stages to
have a proper foundation in the subject.

Not all
students can cope with this type of learning

This
method requires special preparation on the part of the teacher

The
results found out by the student may not be correct. In the absence of
individual checking, all the istakes may not be rectified.

If the
teacher is not able to distinguish between false and true heuristic questions,
he/she may do more harm than good.

Uses

Cultivating a heuristic spirit in students, this
method can be used.

To develop efficiency in problem solving skills,
independent thinking and reasoning, self learning and self-reliance among the
students

Laboratory method

Laboratory method of teaching
mathematics is that method in which we try to make the students learn
mathematics by doing experiments and laboratory work in the mathematics room or
laboratory on the same lines as they learn sciences by performing experiments
in the science rooms or laboratories. It is based on psychological principles
of learning such as ‘learning by doing’, ‘learning by observation’ and so on.
Laboratory method is quite competent to relate the theoretical knowledge with
the practical base. This approach makes the learning process more interesting,
lively and meaningful.

The
success of the laboratory method depends on an able skilled mathematics teacher
as well as the availability of a well-equipped mathematics laboratory.According to J.W.A young “ a room specially
filled with drawing instruments, suitable tables and desks, good black boards
and the apparatus necessary to perform the experiment of the course is really
essential for the best success of the laboratory method”.

Advantages

This method is psychologically more sound as in
it we proceed from known to unknown

This method is based on the principle of
learning by doing

It stimulates the interest of students to work
with concrete material

Knowledge and skills acquired through
experiments help in better understanding and longer retention

It promotes self-confidence and self-reliance
and a sense of achievement among the students

It provides opportunities for social interaction

This method suits the average and below average
students and to such student who cannot understand things at the abstract level

The things learnt experimentally can be used by
the pupil in many situations

Limitations

It
is an expensive method in terms of time, equipment, laboratory facilities and
number of skilled and able teachers

Only
very few topic in mathematics can be taught through this method and hence it
has limited applicability

This
method is not useful for large classes

In
higher classes this method is not suitable. In these classes students are
expected to grasp abstract ideas and concepts

Text
books in use do not indicate the practical work can be done by the students

Uses

It
can be uses in lower classes where we have to make clear certain mathematical
concepts and ideas

Seminar method

The English author Francis Bacon
rightly said long ago, “Reading
makes the full man, writing the exact man and conference the ready man”. That
is the skills of reading, writing and speaking are essential for development of
the personality. These skills are well combined and well developed through the seminar
method. The term ‘seminar’ generally refers to a structured group discussion
that may precede or follow a formal lecture. It may be either in the form of an
essay or a paper presentation. It may be done by experts or academicians at a
seminar or conference organized by an institution or an association or by an
individual student in class or group of peers as a part of the source of work.
In any case, the audience critically examines the paper and discusses the
content or findings of the paper. They make queries for clarification or
elucidation on any point and the writer of the paper is required to clarify
these queries with further facts and figures. The seminar method is very
suitable for higher education or professional discussion when the level of
attainment of the group is relatively high and nature of discussion is expected
to be analytical and technical

Characteristics

The seminar method is stimulating and motivating

It is an effective mode of testing the students
understanding and knowledge in the subject or the topic

It evaluates their skill in arranging,
formulating and presenting facts in a systematic manner

It develops the sense of responsibility and
co-operation as well as the powers of self-reliance and self-confidence

It is an advanced means of socialization and
instruction involving students participation, preparation and interaction

It breaks the monotony of traditional teaching
and promotes academic excellence

It stimulates and tests the students power of
comprehension and understanding

Seminar is mostly subject oriented and deals in
depth a particular topic

Questioning power is developed through
participation

Types of seminar

On the basis of levels of
organization, the seminars are of 4 types (i) Mini seminar (ii) Main seminar
(iii) National seminar and (iv) International seminar

Mini seminar

A seminar organized to discuss a
topic in class is known as mini seminar

Major seminar

Such seminars are organized at
departmental level or institutional level on a major theme

National seminar

A national seminar is organized
by an association or organization at national level. Ex:- NCERT organizes
seminar at national level on different themes like Educational Technology,
Population Education,Distance
Education, Vocational Education etc.

International seminar

Generally such seminars are
organized by UNESCO and other international organization

Limitations

A seminar
cannot be organized on all the content of a subject matter. Some topics are
highly structured

This
technique cannot be used for all levels of education. It can be used for higher
level of education

When a
seminar is being organized, the persons who speak too much, dominate the
discussion of the seminar and do not provide opportunities to take part in the
discussion. It means that the discussion confuses only few persons rather than
whole group

During
discussion, groups are formed in two ideas, anti-ideas and provide on the
theme. As a result they try to win over the other. The purpose of the seminar
is not served

The
opposition group generally try to oppose even for the contractive or relevant
ideas of the other group. This instructional situation of such discussion is
not conducive for learning.

Symposium

Symposium technique is also one
of the techniques of higher learning. It is also an instructional technique
which is used to achieve higher cognitive and affective objectives.

Definitions

“The symposium technique serves
as an excellent device for informing an audience, crystallizing opinion and
generally preparing the listeners for arriving at a decision, policy, value,
judgement or wider understanding”.

The
symposium is a type of discussion, in which two or more speakers talk from ten
to twenty minutes, develop individual approaches or solutions to a problem or
present aspects of apolicy, process or
programme. The speeches are followed by questions or comments from the
audience. The speeches may be persuasive, argumentative, informative or
evocative. Each speech proceeds without interruption. The chairman of the
symposium introduces the topic, suggests something of its importance. Sometimes
indicates the general approaches. The symposium forum serves an excellent
device for informing an audience, crystallizing opinion and in general
preparing the listeners for arriving at a decision policy, value, judgement or
understanding. All members of the performing group can sit in a straight line
behind a table or adjoining chairs, with chair in the middle or to one side of
the speakers or if the symposium is to present two conflicting points of view,
the seating arrangement can separate the speakers on the platform in order to
indicate differences in opinion or in order to preserve peace.

Scope

The symposiumtechnique is used to realize the higher cognitive and affective
objectives

The following are the main topics on which
symposium techniques is used:

Use of television for education

Scope of distance education in our education

Semester system in education

Use of essay and objective type tests

Use of micro teaching in teacher education

Use of team teaching in schools

Characteristics

It provides the broad understanding of a topic
or a problem

The opportunity is provided to the listeners to
take decision about the problem

It is used for higher classes to specific themes
and problems

It develops the feeling of co-operation and
adjustment

The objectives as synthesis and evaluation
(creativity) are achieved by employing the symposium technique.

Limitations

The chairman
has no control over the speakers as they have full freedom to prepare the theme
for discussion

The
discussions and presentation of theme is not summarized at the end. The
participants take decision according to their own. Hence mature persons can
make use of this technique.

There is a
probability of repetition of the conduct because every speaker prepares theme
as a whole. The different aspects of theme are not prepared separately. It
creates difficulty of understanding to the listeners.

Work shop

The workshops are organized to
develop the psychomotor aspects of the learner regarding practices of new
innovations in area of education. Under this technique participants have to do
some practical work to produce instructional, teaching and testing material.

Definitions

Workshop is defined as assembled
group of ten to twenty five persons who share a common interest or problem.
They meet together to improve their individual skill of a subject through
intensive study, research, practice and discussion.

Generally
workshops are organized for three to ten days duration. The period of workshop
may be 40 days, it depends on the nature of task assigned to the workshop. It
is organized in three stages

First stage

Presentation of the theme of
providing awareness

Second stage

Practice the approach for its
applicability

Third stage

Evaluate the material prepared
by the participants and follow-up programme

First stage:

The resource persons or experts
are invited to provide the awareness and understanding of the topic. This stage
is like a seminar type. The paper reading is also alone to discuss the
different aspects of the theme. The trainee or participants are given
opportunities to seek clarification. Stage continues for first two days of
schedule.

Second stage:

In the first stage theoretical
background is provided to the participants. In the second stage the group is
divided in to small groups. Eg: - A workshop for lesson planning or writing
objectives in behavioural terms, question bank preparation etc. The group
formed on the basis of different subjects such as Hindi, English, Maths,
science etc. A resource person or expert provide, guidance and supervision
their work of each trainee of his group. Every trainee has to complete his task
within the given period. At the end they meet in their groups and discuss and
present their task to be completed.

Third stage:

At this stage all groups meet at
one place and present their reports of work done at second stage. The
participants are given opportunities to comment and given suggestions for
further improvement

Follow-up:

The effectiveness of a workshop
is ascertained by a follow up programme. After workshop the trainees have to go
back to their institutions. They are asked to continue their task and examine the
workability and usability in their institution. The participants are invited to
meet again and present their experiences regarding applicability of the topic
or new practices

Scope:

The workshop technique is used
mainly in the following areas of education

1. New format of lesson planning

2. Writing objectives in
behavioural terms

3. Preparing instructional
material or teaching model

4. Workshop for micro-teaching

Advantages

It
is a technique which can be effectively used for developing understanding and
proficiency for the approaches and practices in education.

The
teaching proficiencies can be developed by using the workshop technique for
in-service teachers.

It
provides the opportunities and situations to develop the individual capacities
of a teacher.

It
develops the feeling of co-operation and group work.

The
new practices and innovations are introduced to in-service teachers.

Limitations

The workshops cannot be organized for large
group so that large number of persons are not trained.

The effectiveness of a workshop technique
depends on the follow up programme which does not take place in workshop
technique

Group discussion

There is no comprehensive
definition of group discussion but it is considered as a democratic teaching
strategy. The pupils are more active in it. Teacher’s job is to supervise and
to provide guidance to pupils activities. It is child centered teaching
strategy. It may be of two types

(i) By the teacher and

(ii) By the students

By the teacher: This type of discussion is more autocratic in
style.

By the student: In this situation, discussion is more permissive on
democratic in style.

Objectives: The affective objectives and higher order cognitive
objectives may be achieved.

Structure: The group discussion is organized in two forms: Formally
and informally. In formal group discussion, proper schedule is prepared and
certain rules are observed. In the absence of a teacher pupils have to select a
student to act as a leader of the group. The leader of the group prepares a
plan for discussion. In the group discussion, due weightage is given to the
answer and questions of the pupils.

Principles: This strategy is based up on the following principles
of teaching:

1. Principle of active
participation

2. Principle of freedom for work

3.Principle of group work and
equal opportunities to as questions and to answer them.

Advantages: The following are the main advantages of this strategy

It develops the feeling of group work and group
participation

It helps in developing the creative ability and
thinking among pupils

It develops the problem solving attitude and the
tolerance to hear one’s own criticism.

It helps in developing the feeling of
co-operation

Limitations

It has the following limitations

It has
chances for deviation from the main topic

In the
discussion, only a few students dominate and to discuss more and large number
of students do not participate in the discussion

Generally
groups are formed in this strategy and they are critical to one another.

It
involves more criticism rather than relevant discussion.

Suggestions

It may be
improved and can be effectively employed by observing the following
precautions:

Every
student should be provided an opportunity to participate in the discussion

Students
should be encouraged and motivated to participate in discussion

Irrelevant
criticism should not be encouraged.

Task Analysis

Task analysis is as the name
implies a way of describing how a task is performed. A task analysis involves a
careful examination of the tasks and or objectives in an effort to determine
such things as to how task will be taught, to whom and when. The task analysis
breaks down the task into its constituent parts and involves detailed listing
of component elements of a job or task.

Task
analysis enables us to bring principles desired from learning theory to bear on
the sequencing of instruction. The task analysis assists the teacher to
determine the specific task the student has failed to perform. In task analysis
the knowledge to be taught must be systematically organized so that salient
points are identified.

Techniques of Task analysis

1. Interviews: Individual and
group

2. Direct observation

Role of teacher in task analysis

1. He should administer
diagnostic pre-task before beginning instruction. This enables the teacher to
understand the position of the learner.

2. The teacher should meet the
individual differences of the learners.

3. The teacher must give cues to
the learner which he needs.

Use of computer for teaching of mathematics

In this modern age of computers
through their valuable services and applications represent the most important
gift and contribution of the science and technology to the man kind. The use of
computer in teaching of mathematics may be supported on the following grounds.

1. Diagnosis of the entry behaviour of learners:

Computer programmes can help him
in the early diagnosis of the strength and weaknesses of his students in terms
of their entry behaviour for determining their potentiality.

2. Providing a rich treasure of knowledge and information:

Computers may help in a big way
for the collection, storage, classification and dissemination of knowledge and
information related to the teaching and learning of mathematics. More over
internet provides enormous opportunities for the utilization of information and
knowledge for the attainment of teaching learning objectives in the subject
mathematics.

3. Generating individualized instructional plans:

Depending upon the need,
characteristics, nature and individuality of the learners computer softwares
are able to generate and organize individualized instructional plans for the
benefit of a countless learners at one or the other times at the demand and
pace of the individual learner.

4. Providing valuable teaching aids for instruction:

Computer can help in generating
and producing valuable teaching aids for the teaching and learning of a topic
of a particular unit of concept in the subject mathematics with the use of MS
Office, corel draw, graphics etc we can do the same.

5. Computer help in providing sufficient drill and practice:

Computers can help a lot in
providing sufficient amount of practice and drill work for the fixation of
knowledge and skills already learned. Computer as a machine works quite
instantly and provide proper feedback. Moreover all the students can work here
with their own speed and will to carry on the practice work.

6. Help in doing practical and project work:

Many of the practical skills like
construction work in geometry, surveying and measuring may be well demonstrated
through the well prepared programmes.

7. Providing opportunity for improved self-study:

Computers have the ability and
capacity to provide individual instruction to a limitless number of students at
a single time and there by increase their self learning ability. More over the
student doesn’t remain passive as may happen in case of listening to a
classroom lecture.

8. Monitoring of progress:

Computer may prove quite
effective in monitoring the progress of each and every individual learners in a
quite satisfactory way.

9. Providing remedial instruction:

On the basis of the learning
difficulties diagnosed and the probable causes detected, the softwares can
suggest all the possible remedies helpful in the planning and organization of
remedial instruction to the needed learners.

10. Organisation of testing and evaluation programmes:

Computer softwares may provide
valuable services in the task of managing and organizing testing and evaluation
programmes related to classroom instruction andeducational progress of the
students.

11. Generating all types of reports and records related to the teaching
learning of mathematics:

The data regarding the available
instructional resources, aids and equipments unit wise organization of the
curriculum and the progress regarding to the attainment of instructional
objectives are readily and to the learners for self learning and to the
teachers for imparting appropriate guidance and educational help to the
students. The testing, evaluation and progress reports of the individual
students and the group as a whole can be very well generated through computers.

Importance of teaching aids

There are so many abstractions in
mathematics which cannot be easily followed by the students. To make such
abstract ideas or complex ideas less abstract, teacher takes help of various
teaching aids. Since most of the learning in children takes place at the
sensory level, so the senses are gateways of knowledge. Teaching aids influence
the minds of learners through their senses.

There is a famous saying,

"I hear, I forget

I see, I remember

I do, I understand"

They
help in clear understanding of the subject and clarifying abstract ideas.

The
appeal the senses of the pupils and so they satisfy their innate tendencies and
interests.

They
stimulate pupil participation. Since they are based on the maxim ‘learning by
doing’.

They
make teaching –learning process interesting.

They
help in saving time and energy because it takes a long time to classify an
abstract idea verbally.

Promote
self-activity and purposefulness in learning.

Provide
variety, freshness and reality to learning.

Cater
to individual needs and abilities.

Leave
long lasting impressions on the minds of the learner.

Motivate
the children and stimulate their mental activity.

Projected Aids

A projected aid is one in which
items to be perceived are projected on screen using mechanical devices. The
major projected aids that could be used in teaching are given below.

1. Over head projector (OHP)

This device helps in projecting
transparent images on to a screen under normal light condition.

Uses:

OHP can be used to explain topics and at the
same time teacher can retain eye contact with the students

OHP can be used to avoid the monotony of
continuous audio perception.

2. Episcope and Epidiascope:

Episcope is used for the
projection of opaque objects.

Uses:

It can be used for the projection of pictures,
photographs, drawings or any opaque material with in the age of the platform.

It can be used to enlarge book diagrams and to
illustrate complex materials.

Epidiascope:

Epidiascope is used for the
projection of opaque as well as transparent objects. It is actually a
combination of episcope and diascope.

Uses:

Used for projecting slides

It is possible to produce enlargement of
diagrams, maps or illustrations.

It can be used to reproduce the original colours
of the diagram or picture during projection.

Slide projector

Slide projector is an instrument
equipped with a powerful light source and carrier for holding slides of
suitable size.

Uses:

This can be used when a continuity of a theme is
required.

Slides of different size can be projected.

Film strip projector

Film strip projector are used for
projecting film strips on the screen when a series of slides are to be
presented in a sequence, it is preferable to prepare a film strip.

Graphic aids are visual aids such
as graphs, diagrams, charts etc and are presented on plane surface. The
different categories of graphic aids are described below:

a. Graphs

The nature of the relation of two
dependent variables could be very easily presented by graphical representation.

Uses

It is used to study the
relationship between two variables.

It can be used to get clear
picture and immediate information.

b. Diagrams

Diagrams could be used to explain
many facts at the same time, by using a variety of symbols andlabels.

Use

For getting clear and vivid information at a
glance.

Used for motivating the students

c. Charts

The chart is a systematic
arrangement of key facts or ideas in a logical sequence or representing ideas
and facts in a pictorial or graphic form.

Uses

Motivating the students

Introducing a lesson

Depicting various geometrical figures and their
properties

Comparison of properties of different
geometrical figures

Highlighting key points in a lesson

Concretising abstract mathematical concepts into
visual forms.

(ii) Display boards

The stereo typed atmosphere of
the traditional classroom, libraries and corridors of schools is recently
giving way to attractive and educative surrounding with a wealth of display
materials. A brief description about various types of display boards are given
below.

a. Chalk board

Chalk
board is, perhaps the most important visual aid for teaching mathematics. It is
the most indispensable tool in the hands of a mathematics teacher. Mathematics
cannot be taught effectively without the use of chalk board.

Developing
various steps of the solution of a problem or proof of a theorem.

Writing
important points to draw the attention of the students.

Writing
problems for assignment and drill.

Summarizing
the day’s lesson.

b. Roll-up chalk board

Roll
up chalk board is that which can be conveniently rolled up. These arte usually
made of think rexin cloth.

Uses

The difficulties due to spot
writing or drawing can be solved.

c.Flannel board.

Flannel
board have a fibre surface to which similar materials will adhere.

Uses

Showing the sequence of steps in the derivation
of a formula.

Re arranging the steps in the proper sequence

Moving certain parts of a geometrical figure to
another position so as to get a new geometrical figure.

d. Magnetic board

Magnetic
board is a modification over the flannel board. Instead offlannel, a metallic sheet is fixed on the
plywood. Small magnets are used to hold the materials to be exhibited. It
serves the same purpose as a flannel board does.

3. Three-dimensional aids:

Models

Models are three dimensional
representation of an idea and therefore they are replicas of the original
thing. Models provide contrived experiences where reality is altered or
simplified for teaching purpose. Thus models simplify the reality and enable
the teacher to reduce or enlarge objects to any desirable size.

Uses

For teaching the properties and areas of plane
figures such as cubes, prisms the teacher can make use of these shapes cut out
from chart papers or card board sheets.

While deriving the expansion of algebraic
identities such as ( a + b )3, ( a + b + c )3and so on the teacher can make use of cubes
to get the required terms in the expansion.

4. Audio aids

The
audio aids like radio and tape recorder depend on a single sense – hearing to
convey its rich meanings. When matter is presented through radio and tape
recorder, listening remains the only means of communication.

a. Radio.

Radio
adds to variety to learning experiences. It is a powerful medium for mass
communication.

Uses

It can be used to teach large mass.

It will help the teacher to supplement classroom
instruction

Cost effective and reaches faster to people than
any other device.

b. Tape Recorder

A
tape recorder is used to record sound on magnetic tape which can be reproduced
at will as many times as required

Uses

The recorded lessons by eminent
teachers can be played.

Improvised aids

Improvised aids are the aids
which can be designed and produced from simple articles found in one
surroundings. These aids can be prepared by teachers involving students and
community resources. Some such community resources are sticks, leaves, stones,
shells, broken bangle pieces etc., which are helpful in making learning aids.